The present paper contains the sufficient condition of a fuzzy semigroup to be a fuzzy group using fuzzy points. The existence of a fuzzy kernel in
semigroup is explored. It has been shown that every fuzzy ideal of a semigroup
contains every minimal fuzzy left and every minimal fuzzy right ideal of semigroup.
The fuzzy kernel is the class sum of minimal fuzzy left (right) ideals of a semigroup.
Every fuzzy left ideal of a fuzzy kernel is also a fuzzy left ideal of a semigroup. It
has been shown that the product of minimal fuzzy left ideal and minimal fuzzy
right ideal of a semigroup forms a group. The representation of minimal fuzzy left
(right) ideals and also the representation of intersection of minimal fuzzy left ideal
and minimal fuzzy right ideal are shown. The fuzzy kernel of a semigroup is basically
the class sum of all the minimal fuzzy left (right) ideals of a semigroup. Finally
the sufficient condition of fuzzy kernel to be completely simple semigroup has been
proved.

1. Introduction

Zadeh in 1965 introduced the fundamental concept of a fuzzy set in his paper [1] which provides a useful mathematical tool for describing the behavior of systems that are too complex or ill-defined to admit precise mathematical analysis by classical methods. The literature in fuzzy set theory and its practicability has been functioning quickly uptil now. The applications of these concepts can now be seen in a variety of disciplines like artificial intellegence, computer science, control engineering, expert systems, operation research, management science, and robotics.

Mordeson [2] has demonstrated the basic exploration of fuzzy semigroups. He also gave a theoratical exposition of fuzzy semigroups and their application in fuzzy coding, fuzzy finite state machines, and fuzzy languages. The role of fuzzy theory in automata and daily language has extensively been discussed in [2].

Fuzzy sets are considered with respect to a nonempty set S. The main idea is that each element x of S is assigned a membership grade f(x) in [0,1], with f(x)=0 corresponding to nonmembership, 0<f(x)<1 to partial membership, and f(x)=1 to full membership. Mathematically, a fuzzy subset f of a set S is a function from S into the closed interval [0,1]. The fuzzy theory has provided generalization in many fields of mathematics such as algebra, topology, differential equation, logic, and set theory. The fuzzy theory has been studied in the structure of groups and groupoids by Rosenfeld [3], where he has defined the fuzzy subgroupoid and the fuzzy left (right, two-sided) ideals of a groupoid S. He has used the characteristic mapping(1)CA={1ifx∈A0ifx∉A
of a subset A of a groupoid S to show that A is subgroupoid or left (right, two-sided) ideal of S if and only if CA is the fuzzy subgroupoid or fuzzy left (right, two-sided) ideal of S. Fuzzy semigroups were first considered by Kuroki [4] in which he studied the bi-ideals in semigroups. He also considered the fuzzy interior ideal of a semigroup, characterization of groups, union of groups, and semilattices of groups by means of fuzzy bi-ideals. N. Kehayopulu, X. Y. Xie, and M. Tsingelis have produced several papers on fuzzy ideals in semigroup.

A nonempty subset A of a semigroup S is a left (right) ideal of S if SA⊆A(AS⊆A) and is ideal if A is both left and right ideal of S. An ideal A of S is called minimal ideal of S if A does not properly contains any other ideal of S. If the intersection K of all the ideals of a semigroup S is nonempty then we shall call K the kernel of S. Huntington in [5] has shown the simplified definition of a semigroup to be a group. We have used this concept in fuzzy semigroups and have shown that the product of minimal fuzzy left ideal and minimal fuzzy right ideal forms a group. Clifford in [6] had given the representation of minimal left (right) ideal of a semigroup. We have used this ideas to extend the fuzzy ideal theory of minimal fuzzy ideal in semigroups and have given a representation of fuzzy kernel in terms of minimal fuzzy left (right) ideals of a semigroup.

For a semigroup S, F(S) will denote the collection of all fuzzy subsets of S. Let f and g be two fuzzy subsets of a semigroup S. The operation “∘” in F(S) is defined by;
(2)(fog)(x)={⋁x=yz{f(y)∧g(z)},if∃y,z∈S,suchthatx=yz,0,otherwise.
For simplicity, we will denote F(S) as S.

2. Sufficient Condition of a Semigroup to Be a Group

The very first definition of fuzzy point is given by Pu and Liu [7]. Let A be a nonempty set and λ∈[0,1]. A fuzzy point aλ of A means that aλ is a fuzzy subset of A defined by
(3)aλ(x)={λ,ifx=a,0,ifx≠a.

Each fuzzy point aλ of A is the fuzzy subset of a set A [7]. Every fuzzy set A can be expressed as the union of all the fuzzy points in A. It is to be noted that for any x and y in A, λ, and μ in [0,1] the fuzzy points xλ and yμ have the inclusion relation that is xλ⊆yμ if and only if x=y and λ≤μ.

In the following text for λ,μ,t,γ,δ,α,β, and ρ in [0,1], we denote aλ,bμ,xt,xμ,eα,fβ, and cρ as the fuzzy points of different fuzzy subsets of S.

Theorem 1.

If in a semigroup S, the following conditions hold:(i)

for any fuzzy points aλ and bμ of S there exists xγ in S, such that aλ∘xγ=bμ,

(ii)

for any fuzzy points aλ and bμ of S there exists yγ in S, such that yγ∘aλ=bμ, then (S,∘) is a fuzzy group.

Proof.

First of all we will show that the fuzzy points xγ and yγ existing in the hypothesis are uniquely determined. To show that xγ in condition (i) of the theorem is unique, let us assume on contrary that there exists two fuzzy points xγ and zδ such that, aλ∘xγ=bμ and aλ∘zδ=bμ. Thus,
(4)aλ∘xγ=aλ∘zδ.
The condition (ii) in the theorem implies that for the fuzzy points xγ and zδ of S there exists a fuzzy point cγ of S such that zδ∘cγ=xγ. Hence (4) implies that,
(5)aλ∘(zδ∘cγ)=aλ∘zδ,(6)(aλ∘zδ)∘cγ=aλ∘zδ.
Now, we will show that if for any fuzzy point aλ of S,aλ∘xγ=aλ then for each fuzzy point bμ of S, bμ∘xγ=bμ. By the condition (ii) for fuzzy points aλ and bμ of S there exists a fuzzy point yλ of S such that yγ∘aλ=bμ. Now by hypothesis,
(7)yγ∘(xγ∘aλ)=bμ,(yγ∘xγ)∘aλ=bμ,bμ∘aλ=bμ.
Therefore (6) implies that zδ∘cγ=zδ, and so xγ=zδ. It has been shown that the left cancellative law holds in S.

Similarly, we can show that for any fuzzy points aλ and bμ of S there exists a unique fuzzy point yγ in S, such that yγ∘aλ=bμ. Similarly, we can show that S is a right cancellative.

Hence the cancellation law holds in a semigroup S and thus for each fuzzy point aλ of S we have, aλ∘S=S and S∘aλ=S. Now there exists fuzzy points eα and fβ of S such that fβ∘aλ=aλ and aλ∘eα=aλ. Let xγ be any arbitrary fuzzy point of S then xγ is in S∘aλ, that is there exists bμ in S such that xγ=bμ∘aλ. Thus, xγ∘eα=(bμ∘aλ)∘eα=bμ∘(aλ∘eα)=bμ∘aλ=xγ, which implies that eα is the right identity in S. Similarly we can show that fβ is the left identity in S. Now since fβ is the left identity so fβ∘eα=eα and eα is the right identity so fβ∘eα=fβ. Hence fβ=eα, which implies that S has an identity element eα in S. Now for eα in S, there exists xγ and yρ in S such that xγ∘aλ=eα and aλ∘yρ=eα. Now
(8)(xγ∘aλ)∘yρ=eα∘yρ,xγ∘(aλ∘yρ)=yρ,xγ∘eα=yρ,xγ=yρ.

Hence there exists an inverse element for each fuzzy point of S. The uniqueness of identity and inverse elements can be proved easily.

The following corollary can be easily asserted from the proof of Theorem 1.

Corollary 2.

A fuzzy semigroup S is a fuzzy group if and only if aλ∘S=S and S∘aλ=S for all fuzzy points aλ of S.

3. Minimal Fuzzy Ideals of a Semigroup

The nonempty intersection of fuzzy ideals of semigroup S is called the fuzzy kernel of a semigroup S. The fuzzy ideal f of S is called minimal fuzzy ideal if f contains no proper fuzzy ideal of S. A semigroup S is called a fuzzy simple if it does not contain any proper fuzzy ideal of S.

The following lemmas are in our previous knowledge [2].

Lemma 3.

For any nonempty subsets A and B of semigroup S, we have A⊆B if and only if CA⊆CB.

Lemma 4.

Let A be a nonempty subset of a semigroup S, then A is an ideal of S if and only if CA is a fuzzy ideal of S.

Lemma 5.

Let f and g be fuzzy ideals of a semigroup S, then f∘g and g∘f are also fuzzy ideals of S.

Lemma 6.

Let f and g be fuzzy ideals of a semigroup S, then f∩g and g∩f are also fuzzy ideals of S.

Theorem 7.

A nonempty subset A of a semigroup S is minimal ideal if and only if CA is minimal fuzzy ideal of S.

Proof.

Let A be a minimal ideal of S, then by Lemma 4, CA is a fuzzy ideal of S. Suppose that CA is not minimal fuzzy ideal of S, then there exists some fuzzy ideal CB of S such that, CB⊆CA. Hence by Lemma 3, B⊆A, where B is an ideal of S. This is a contradiction to the fact that A is minimal ideal of S. Thus CA is the minimal fuzzy ideal of S. Conversely, let CA be the minimal fuzzy ideal of S, then by Lemma 4, A is the ideal of S. Suppose that A is not minimal ideal of S, then there exists some ideal B of S such that B⊆A. Now by Lemma 3, CB⊆CA, where CB is the fuzzy ideal of S. This is a contradiction to the fact that CA is the minimal fuzzy ideal of S. Thus A is the minimal ideal of S.

Theorem 8.

If a semigroup S contains a minimal fuzzy ideal f of S then f is the fuzzy kernel of S.

Proof.

Let g be any fuzzy ideal of S then for a minimal fuzzy ideal f of S,g∘f⊆g∩f. Thus g∩f is non-empty. Since by Lemma 6, g∩f is a fuzzy ideal of S and g∩f⊆f, which implies that g∩f=f. But then f=g∩f⊆g, so f contains in every fuzzy ideal of S and hence is a fuzzy kernel of S.

Theorem 9.

If a semigroup has fuzzy kernel f then f is a simple subsemigroup of S.

Proof.

Since f is a fuzzy ideal of S, so f is a fuzzy subsemigroup of S, since f∘f⊆S∘f⊆f. To show f is simple, let g be any fuzzy ideal of f, then f∘g∘f is fuzzy ideal of S, since S∘(f∘g∘f)=(S∘f)∘g∘f⊆f∘g∘f and (f∘g∘f)∘S=f∘g∘(f∘S)⊆f∘g∘f. Also f∘g∘f⊆f∘f⊆f, but by Theorem 8, f is minimal fuzzy ideal of S because every fuzzy kernel of S, if exists, is a minimal fuzzy ideal of S. Hence f∘g∘f=f. Also f∘g∘f⊆g∘f⊆g, which implies that f⊆g. Thus f=g, implies that f is simple subsemigroup of S.

Lemma 10.

If f is a minimal left ideal of a semigroup S and cλ be any fuzzy point of S for λ in [0,1] then f∘cλ is also a minimal fuzzy left ideal of S.

Proof.

Since for a minimal fuzzy left ideal f of S,S∘(f∘cλ)=(S∘f)∘cλ⊆f∘cλ, which implies that f∘cλ is a fuzzy left ideal of S. Suppose g is the fuzzy left ideal of f∘cλ and let h={aμ∈f:aμ∘cλ⊆gforμ∈[0,1]}. Let for t in [0,1],bt be a fuzzy point of S then for aμ∘cλ in h,bt∘aμ is in f and so bt∘aμ∘cλ⊆g. Now bt∘aμ⊆h, which implies that S∘h⊆h. Hence h is fuzzy left ideal of S contained in minimal fuzzy left ideal f of S and so h=f. Thus for all aμ in f,aμ∘cλ⊆g, which implies that f∘cλ⊆g. Hence f∘cλ=g, and so f∘cλ is a minimal fuzzy left ideal of S.

Lemma 11.

A fuzzy ideal f of a semigroup S contains every minimal fuzzy left ideal of S.

Proof.

Let g be any minimal fuzzy left ideal of S, then f∘g is fuzzy left ideal of S since, S∘(f∘g)=(S∘f)∘g⊆f∘g. Also f∘g⊆g as g is a fuzzy left ideal of S. But g is minimal so g=f∘g⊆f. Hence every minimal fuzzy left ideal g of S is contained in every fuzzy ideal of S.

Theorem 12.

If a semigroup S contains at least one minimal fuzzy left ideal then it has a fuzzy kernel of S, where fuzzy kernel is the class sum of all the minimal fuzzy left ideals of S.

Proof.

Let f be the union of all the minimal fuzzy left ideals of S. Since S contains at least one minimal fuzzy left ideal of S, so f is non-empty. Let f1,f2,f3,… be all the minimal fuzzy left ideals of S then, S∘f=S∘(f1∪f2∪f3∪⋯)⊆S∘f1∪S∘f2∪S∘f3∪⋯⊆f1∪f2∪f3∪⋯=f, which implies that f is the fuzzy left ideal of S. Now it has to be shown that f is minimal. Let, for λ,μ in [0,1],aλ and bμ be arbitrary fuzzy points of S and f, respectively. By definition of f, the fuzzy point bμ is in some fi for some i∈ℕ, such that bμ∘aλ⊆fi∘aλ. Now since fi is minimal fuzzy left ideal of S so by Lemma 10, fi∘aλ is also minimal fuzzy left ideal of S and hence contained in f. Thus, bμ∘aλ⊆f and so f∘S⊆f. Hence f is fuzzy right ideal of S and so fuzzy ideal of S. Now by Lemma 11, every fuzzy ideal contains every minimal fuzzy left ideal of S and so f being union of all minimal fuzzy left ideals of S is contained in every fuzzy ideal. Hence f is the fuzzy kernel of S.

Theorem 13.

Let a semigroup S contain at least one minimal left ideal then every fuzzy left ideal of fuzzy kernel is also a fuzzy left ideal of S.

Proof.

Let f be the fuzzy kernel of S, then by Theorem 12, f is the class sum of all the minimal fuzzy left ideals of S. Let g be a fuzzy left ideal of f, then each fuzzy point aλ for λ in [0,1], of f belongs to some minimal fuzzy left ideal fi,i∈ℕ of S. Now f∘aλ is a fuzzy left ideal of S, since S∘(f∘aλ)=(S∘f)∘aλ⊆f∘aλ. Also f∘aλ⊆f∘fi⊆fi, where fi is minimal fuzzy left ideal so f∘aλ=fi. Thus aλ⊆f∘aλ implies that f⊆f∘aλ, where f∘aλ is minimal fuzzy left ideal of S and so f=f∘aλ. Hence, f is the fuzzy left ideal of S.

Remark 14.

Every minimal fuzzy left ideal of S is a minimal fuzzy left ideal of fuzzy kernel of S and vice versa.

Theorem 15.

Let a semigroup S contain at least one minimal fuzzy left ideal of S then every fuzzy left ideal of S contains at least one minimal fuzzy left ideal of S.

Proof.

Let f be the fuzzy left ideal of S, then by Theorem 12, S has a fuzzy kernel g, where g is the class sum of all the minimal left ideals of S. Now g∘f is a fuzzy left ideal of S, since S∘(g∘f)=(S∘g)∘f⊆g∘f, and also g∘f⊆g. Thus g∘f is one of the minimal fuzzy left ideals of S or contains some minimal fuzzy left ideals of S. But g∘f is also contained in f. Hence f contains at least one minimal fuzzy left ideal of S.

Remark 16.

Lemmas 10 and 11 and Theorems 12, 13, and 15 are also applicable for fuzzy right ideal of a semigroup.

Theorem 17.

If a semigroup S contains at least one minimal fuzzy left ideal and one minimal fuzzy right ideal of S, then the class sum of all the minimal fuzzy left ideals of S coincide with the class sum of all the minimal fuzzy right ideals and constitutes the fuzzy kernel of S.

Proof.

By Theorem 12 the class sum of all the minimal fuzzy left ideals of S is a fuzzy kernel of S. Similarly the class sum of all the minimal fuzzy right ideals of S is also the fuzzy kernel of S. But fuzzy kernel of a semigroup S is unique since it is the intersection of all the fuzzy ideals of S.

From now on we assume that fR and fL will denote the minimal fuzzy right ideal and the minimal fuzzy left ideal of a semigroup S, respectively. A fuzzy idempotent point xλ of semigroup S is said to be under another fuzzy idempotent point yμ if xλ∘yμ=yμ∘xλ=xλ. The fuzzy idempotent xλ is called fuzzy primitive if there is no fuzzy idempotent under xλ. A simple fuzzy semigroup S is said to be completely simple if every fuzzy idempotent point of S is fuzzy primitive and for each fuzzy point aγ of S there exists fuzzy idempotents xλ and yμ such that xλ∘aγ=aγ∘yμ=aγ.

Lemma 18.

Let aλ be in fR and bμ be in fR∘fL, then there exists a fuzzy point xt in fR∘fL such that aλ∘xt=bμ.

Proof.

Note that aλ∘fR is a fuzzy right ideal of S since, (aλ∘fR)∘S=aλ∘(fR∘S)⊆aλ∘fR and also aλ∘fR⊆fR∘fR⊆fR. But fR is the minimal fuzzy right ideal of S so aλ∘fR=fR. Hence aλ∘fR∘fL=fR∘fL.

Lemma 19.

Let aλ be in fL and bμ be in fR∘fL, then there exists a fuzzy point xt in fR∘fL such that xt∘aλ=bμ.

Proof.

Since fL∘aλ is a fuzzy left ideal of S since, S∘(fL∘aλ)=(S∘fL)∘aλ⊆fL∘aλ and also fL∘aλ⊆fL∘fL⊆fL. But fL is the minimal fuzzy left ideal of S so fL∘aλ=fL. Hence fR∘fL∘aλ=fR∘fL.

Theorem 20.

fR∘fL is a group.

Proof.

Since (fR∘fL)∘(fR∘fL)⊆(fR∘fL)∘fL⊆fR∘fL, which implies that fR∘fL is a semigroup. Let aλ and bμ be any two fuzzy points of fR∘fL then aλ and bμ are in both fR and fL, since fR∘fL⊆fR and fR∘fL⊆fL. Thus by Lemmas 18 and 19 there exists xγ and yρ in S such that, aλ∘xγ=bμ and yρ∘aλ=bμ. Hence by Theorem 1, fR∘fL is a group.

Lemma 21.

Let eα be the identity element of the group (fR∘fL,∘), then fR=eα∘S,fL=S∘eα and fR∩fL=eα∘S∘eα.

Proof.

Since identity element is in fR since fR∘fL⊆fR. Now (eλ∘S)∘S=eλ∘(S∘S)⊆eλ∘S, which implies that eλ∘S is a fuzzy right ideal of S and is contained in minimal fuzzy right ideal fR of S and hence eλ∘S=fR. Similarly we can show that S∘eλ=fL. Now fR∘fL=(eλ∘S)∘(S∘eλ)=eλ∘(S∘S)∘eλ⊆eλ∘S∘eλ, also eλ∘S∘eλ⊆fR∘S∘fL⊆fR∘fL.

Theorem 22.

For minimal fuzzy right ideal fR and minimal fuzzy left ideal fL of a semigroup S,fR∘fL=fR∩fL.

Proof.

We only need to show that fR∩fL⊆fR∘fL. Let aλ be arbitrary fuzzy point of fR∩fL and let eα be identity element of fR∘fL then by Lemma 18 there exists fuzzy point xγ of fR∘fL such that aλ∘xγ=eα. Let yρ be the inverse fuzzy element of xγ in fR∘fL then, aλ∘eα=aλ∘(xγ∘yρ)=(aλ∘xγ)∘yρ=eα∘yρ=yρ. Now aλ⊆fR∘fL⊆fL, so by Lemma 21, aλ∘eα=eα. Hence aλ=yρ and so aλ is in fR∘fL.

Lemma 23.

The identity fuzzy point eλ of the fuzzy group fR∘fL is a fuzzy primitive.

Proof.

Let xμ be a fuzzy idempotent under eλ, then xμ∘eλ=eλ∘xμ=xμ. Further, Lemma 21 implies that xμ is in fR∩fL. By Theorem 22xμ is in the fuzzy group fR∘fL, which can contain only one fuzzy idempotent point, namely the identity element eλ, so xμ=eλ.

Theorem 24.

Let a semigroup S have at least one minimal fuzzy left ideal and at least one minimal fuzzy right ideal of S then the fuzzy kernel fK of S is completely simple semigroup.

Proof.

Every fuzzy kernel of a semigroup is simple. Each fuzzy point aμ of the fuzzy kernel fK of S belongs to exactly one minimal fuzzy left ideal fL and to exactly one minimal fuzzy right ideal fR of S, otherwise fL and fR are not the minimal fuzzy left and minimal fuzzy right ideals of S. So the fuzzy point aμ belongs to unique fuzzy group fR∘fL. Thus fK is the union of the disjoint fuzzy groups fR∘fL. Each fuzzy idempotent point of fK must be in one of these fuzzy groups, which can only have the fuzzy idempotent point eλ, namely the identity point. But by Lemma 23 the identity point is fuzzy primitive. The second condition for fK to be completely simple is straightforward as each fuzzy point aμ of fK is in one of the fuzzy groups fR∘fL having fuzzy idempotent eλ such that eλ∘aμ=aμ∘eλ=aμ. Hence fK is completely simple semigroup.